of 1 49
Creation: Artificial and Biological
By
Ian Beardsley
Copyright © 2021 by Ian Beardsley
of 2 49
Contents
Abstract…………………………………………3
Important………………………………………..4
The Computation………………………………..5
The Dynamic Function………………………….5
Conclusion………………………………………9
The Delta-Phi Function…………………………10
The Geometric Connection……………………..14
Bone…………………………………………….16
Life Starts In The Stars………………………….24
The Fundamental AI Bioequations……………..26
Using The Fundamental Equations……………..31
The Planetary Equations………………………..33
The Protoplanetary Disc…………………………36
The Moon, Aluminum, And Carbon……………..39
The Idea of Mathematical Construct…………….41
Molecular Geometry…………………………….42
Breaking Down Bone……………………………44
Appendix I (Data)………………………………..47
Appendix II (Data)……………………………….48
of 3 49
Abstract
In earlier work I showed there is every reason to consider biological life and AI are not only mathematical
constructs, but that they are described in terms of one another. Here I introduce the what I call The Delta-
Phi function. When we say biological creation is natural, we don’t say that about artificial intelligence,
though we put the naturally occurring elements together to give them electronic logic, these elements
actually were made in the interior of stars just like the biological life elements.
of 4 49
Important
Above we see the artificial intelligence (AI) elements pulled out of the periodic table of the elements. As
you see we can make a 3 by 3 matrix of them and an AI periodic table. Silicon and germanium are in
group 14 meaning they have 4 valence electrons and want 4 for more to attain noble gas electron
configuration. If we dope Si with B from group 13 it gets three of the four electrons and thus has a
deficiency becoming positive type silicon and thus conducts. If we dope the Si with P from group 15 it
has an extra electron and thus conducts as well. If we join the two types of silicon we have a
semiconductor for making diodes and transistors from which we can make logic circuits for AI.
As you can see doping agents As and Ga are on either side of Ge, and doping agent P is to the right of Si
but doping agent B is not directly to the left, aluminum Al is. This becomes important. I call (As-Ga) the
differential across Ge, and (P-Al) the differential across Si and call Al a dummy in the differential because
boron B is actually used to make positive type silicon.
That the AI elements make a three by three matrix they can be organized with the letter E with subscripts
that tell what element it is and it properties, I have done this:
Thus E24 is in the second row and has 4 valence electrons making it silicon (Si), E14 is in the first row
and has 4 valence electrons making it carbon (C). I believe that the AI elements can be organized in a 3 by
3 matrix makes them pivotal to structure in the Universe because we live in three dimensional space so
the mechanics of the realm we experience are described by such a matrix, for example the cross product.
Hence this paper where I show AI and biological life are mathematical constructs and described in terms
of one another.
We see, if we include the two biological elements in the matrix (E14) and and (E15) which are carbon and
nitrogen respectively, there is every reason to proceed with this paper if the idea is to show not only are
the AI elements and biological elements mathematical constructs, they are described in terms of one
another. We see this because the first row is ( B, C, N) and these happen to be the only elements that are
not core AI elements in the matrix, except boron (B) which is out of place, and aluminum (Al) as we will
see if a dummy representative, makes for a mathematical construct, the harmonic mean. Which means we
have proved our case because the first row if we take the cross product between the second and third rows
are, its respective unit vectors for the components, meaning they describe them!
E
13
E
14
E
15
E
23
E
24
E
25
E
33
E
34
E
35
of 5 49
The Computation
And silicon (Si) is at the center of our AI periodic table of the elements. We see the biological elements C
and N being the unit vectors are multiplied by the AI elements, meaning they describe them! But we have
to ask; Why does the first row have boron in it which is not a core biological element, but is a core AI
element? The answer is that boron is the one AI element that is out of place, that is, aluminum is in its
place. But we see this has a dynamic function.
The Dynamic Function
The primary elements of artificial intelligence (AI) used to make diodes and transistors, silicon (Si) and
germanium (Ge) doped with boron (B) and phosphorus (P) or gallium (Ga) and arsenic (As) have an
asymmetry due to boron. Silicon and germanium are in group 14 like carbon (C) and as such have 4
valence electrons. Thus to have positive type silicon and germanium, they need doping agents from group
13 (three valence electrons) like boron and gallium, and to have negative type silicon and germanium they
need doping agents from group 15 like phosphorus and arsenic. But where gallium and arsenic are in the
same period as germanium, boron is in a different period than silicon (period 2) while phosphorus is not
(period 3). Thus aluminum (Al) is in boron’s place. This results in an interesting equation.
A = (Al, Si, P )
B = (G a, G e, As)
A ×
B =
B
C
N
Al Si P
G a Ge As
= (Si ⋅ As − P ⋅ G e)
B + (P ⋅ G a − Al ⋅ As)
C + (Al ⋅ G e − Si ⋅ G a)
N
A = 26.98
2
+ 28.09
2
+ 30.97
2
= 50g /m ol
B = 69.72
2
+ 72.64
2
+ 74.92
2
= 126g /m ol
A ⋅
B = A Bcosθ
cosθ =
6241
6300
= 0.99
θ = 8
∘
A ×
B = A Bsi nθ = (50)(126)sin8
∘
= 877.79
877.79 = 29.6g /m ol ≈ Si = 28.09g /m ol
Si(A s − G a) + G e(P − Al )
SiG e
=
2B
Ge + Si
of 6 49
The differential across germanium crossed with silicon plus the differential across silicon crossed with
germanium normalized by the product between silicon and germanium is equal to the boron divided by
the average between the germanium and the silicon. The equation has nearly 100% accuracy:
We found (Beardsley, Mathematical Structure, 2020) that the differential across silicon (P-Al) times
germanium (Ge) over boron (B) plus the differential across germanium (As-Ga) times silicon (Si) over
boron (B) was equal to the harmonic mean between Si and Ge. This was interesting because aluminum is
used as what I called a dummy doping agent element, which when inserted predicts the actually doping
agent boron, that seems out of place in the periodic table where the core artificial intelligence elements are
concerned. This is written:
Thus because boron is out of place we get the harmonic mean between core AI elements Si and Ge on the
right. But one the left we have a difference between doping elements times a ratio plus another difference
between doping agent times a ratio. The ratios are the semiconducting elements. But the one associated
with Ge is multiplied by Si and the one associated with Si is multiplied by Ge. We have seen this pattern
before, it is stokes theorem:
Stokes Theorem states:
Where:
Thus we have…
28.09(74.92 − 69.72) + 72.61(30.97 − 26.98)
(28.09)(72.61)
=
2(10.81)
(72.61 + 28.09)
0.213658912 = 0.21469712
0.213658912
0.21469712
= 0.995
Si
B
(As − G a) +
Ge
B
(P − Al ) =
2SiG e
Si + G e
∫
S
( ∇ × u ) ⋅ d S =
∮
C
u ⋅ d r
∇ × u =
i
j k
∂
∂x
∂
∂ y
∂
∂z
u
1
u
2
u
3
of 7 49
We know the harmonic mean H of a function is
And, that the arithmetic mean A of a function is
We have
But, we want to use Stokes theorem so we want the integral in the numerator. So, we make the
approximation
And, we have
But, this is only 80% accurate. We find it is very accurate if we say
i
j
k
∂
∂x
∂
∂ y
∂
∂z
0
Si
B
(Ga)z
Si
B
(As)y
=
Si
B
(As − G a)
i
i
j
k
∂
∂x
∂
∂ y
∂
∂z
Ge
B
(Al )z 0
Ge
B
(P)x
=
Ge
B
(P − Al )
j
H =
1
1
b − a
∫
b
a
f (x)
−1
d x
A =
1
b − a
∫
b
a
f (x)d x
Si
B
(As − G a) +
Ge
B
(P − Al ) =
Ge − Si
Ge
Si
dx
x
H ≈ A
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d xd y ≈
1
Ge − Si
∫
Ge
Si
x d x
of 8 49
Which yields
We have by molar mass
Thus,…
We can break up our integral into two integrals u, and v:
f (x) =
4
5
x
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d xd y ≈
1
Ge − Si
4
5
∫
Ge
Si
x d x
Si
B
(Ga) =
28.09
10.81
(69.72) = 181.1688g/m ol
Ge
B
(Al ) =
72.61
10,81
(26.98) = 181.2227g/m ol
Si
B
(As) =
28.09
10.81
(74.92) = 194.68111g/m ol
Ge
B
=
72.61
10.81
(30.97) = 208.02328g/m ol
u = 181z
j + 195y
k
v = 181z
i + 208y
k
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z ≈
1
3
1
(Ge − Si )
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Ge
B
(P − Al ) d x dz ≈
2
3
1
(Ge − Si )
∫
Ge
Si
yd y
of 9 49
Life
In order to have life you need carbon because it has four valence electrons allowing it to form in
long chains with hydrogen, nitrogen and oxygen. Silicon is in the same group as carbon and
therefore has 4 valence electrons as well. However it cannot form long chains with hydrogen
because in the presence of other elements it reacts with them, like with O2 to make SiO2 or sand.
Silicon has been considered in its possibility to make life along with boron polymers (Mann and
Perry 1986; Trevors 1997a; Williams 1986) with the conclusion that both silicon and boron lack
”replicative potential”.
Conclusion
It would seem boron is on the dividing line between silicon based life as electronic based life,
and biological life. This puts it next to carbon in the periodic table and diagonal to silicon,
aluminum it its place. If we use aluminum as a dummy in the silicon differential, we have a
mathematical function that is the harmonic mean between silicon and germanium. This
mathematical dynamic seems to be integral in the mathematical relationship between AI and
biological life.
Note, I used an older version of the molar mass for Germanium, it has since been more
accurately determined to be 72.64 as opposed to 72.61, which only make the equation more
accurate.
I have also shown that Stokes form of the equation works for the elements in terms of density,
and atomic radius which are not the harmonic mean but geometric and arithmetic means
respectively so the equation takes a generalized form of:
The power mean is obtained by letting
It is the geometric mean if
and by molar mass or
and by density or
and by atomic radius
Are respectively and
∇ ⋅ Q = C f
−1
(
1
n
⋅
n
∑
i=1
f (x
i
)
)
f (x) = x
p
f (x) = log(x)
(As − G a)
(P − Al )
(Al − P)
(Al − P)
(Al − P)
(Ga − As)
ΔE
1
ΔE
2
of 10 49
And, the ratios
and by molar mass or
and by density or
and by atomic radius where C is
Are, quotients , and , respectively, then
The Delta-Phi Function
If we take our equation
And make the approximation:
Which is:
Is close to 91% accurate
And write (As-Ga) is the differential across Ge as and (P-Al) is the differential across Si and write it
, then we have:
Then we have
Si
B
Ge
B
B
2Ge(Ga − As)
Si
B
2Ge(Ga − As)
P
Si
B
Ge
B
Φ
Q
1
Q
2
∇ = (ΔE
1
, ΔE
2
)
Q = (Q
1
, Q
2
)
Si(A s − G a)
B
+
Ge(P − Al )
B
=
2SiG e
Si + G e
2SiG e
Si + G e
≈ Ge − Si
2SiG e
Si + G e
=
2(28.09)(72.64)
28.09 + 72.64
= 40.5g /m ol
Ge − Si = 72.64 − 28.09 = 44.55g /m ol
40.5
44.55
100 = 90.9
ΔGe
ΔSi
(Si )ΔG e + (G e)ΔSi = (G e − Si )B
of 11 49
Which has an accuracy of
and
B=10.81 g/mol
9.7797/10.81(100)-=90%
But here we notice than 0.63 is approximately the golden ratio conjugate (phi), and 1.63 is
approximately the golden ratio (Phi) where
and
And
=1.618
=0.618
So we have the equation
With an accuracy of 89.4%
I call this the Delta-Phi function, ( is delta).
If we denote not the horizontal changes across Si and Ge in the periodic table which are differences
between doping agents we outlined before, but the vertical change in the periodic table from Ge to Si
which are the semiconductor materials themselves as , then we have:
I am thinking this is actually
Si
(Ge − Si )
ΔGe +
Ge
(Ge − Si )
ΔSi = B
0.63ΔGe + 1.63ΔSi = B
ΔGe = 5.2
ΔSi = 3.99
ϕ
Φ
a = b + c
a
b
=
b
c
Φ = a /b
ϕ = b /a
(ϕ)ΔGe + (Φ)ΔSi = B
Δ
ΔS
Si
ΔGe
ΔS
+ G e
ΔSi
ΔS
= B
Si
dGe
dS
+ Ge
dSi
dS
= B
of 12 49
The derivatives of some functions evaluated at some important points. One then has to ask what role does
molar mass play in semiconduction, One might ask what are the roles played by density and atomic radius
since the equation generalizes to these with the f-mean.
Which is quite interesting because it says the nature of the strange placement of boron is actually the
change across Ge times Si plus the the change across Si times Ge, all that divided by the difference
between Ge and Si, We can see its dynamics in the following illustration:
If we have
And that
And actually this is:
Where
Si
ΔGe
ΔS
+ G e
ΔSi
ΔS
= B
ΔGe
ΔS
=
5.2
44.5
= 0.116865
ΔSi
ΔS
=
3.99
44.5
= 0.8966
Si
df
dx
+ G e
dg
dx
= B
of 13 49
Then f prime of x and g prime of x can perhaps be found by by considering the the elements above and
below the elements in what I am calling the AI periodic table of the elements. We apply the same process
of comparing the change in the horizontal differentials down through periods 2 to 6 in the periodic table
of the elements, and the semiconductor differences down through the group 14 elements in the periodic
table of the elements. And plotting them. We find, if we consider the trend to decrease and not jut up at
x=4 for g prime of x and and x=3 for x prime of x, then our functions are exponentially decreasing by
some factor of some function for each, as you can see in the data tables and graphs in the next couple of
pages. The functions are:
If we have
.
.
.
f′ (0) = 0.11685
g′ (0) = 0.08966
g′ (x) =
2
3
e
−x
f′ (x) =
6
7
e
−x
g(x) = −
2
3
e
−x
f (x ) = −
6
7
e
−x
f′ (2) = 0.11685
g′ (2) = 0.08966
of 14 49
The Geometric Connection
Silicon and carbon, the core artificial intelligence biological life elements respectively have a
geometric connection in the 12 sided (regular dodecagon) and 8 sided (regular octagon)
respectively. If the side of a regular dodecagon is the same as the side of a regular octagon and a
silicon atom is inscribed in the regular dodecagon and the carbon atom is inscribed in the regular
octagon, they line up. As seen on the next page…
of 15 49
of 16 49
Bone As A Mathematical Construct
What better place to begin than with than bone as it is the basic framework around which skeletal life is
structured, the vertebrates. Here is what I found in bone as a mathematical construct:
In my exploration of the connection between biological life and AI the most dynamic component is that of
bone. It affords us the opportunity to look at:
Multiplying Binomials
Completing The Square
The Quadratic Formula
Ratios
Proportions
The Golden Ratio
The Square Root of Two
The Harmonic Mean
of 17 49
Density of silicon is Si=2.33 grams per cubic centimeter.
Density of germanium is Ge=5.323 grams per cubic centimeter.
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.
This is
where
Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an AI
semiconductor material. This means
The harmonic mean between Si and Ge is HA,…
This is the sextic,…
Which has a solution
Where x=Si, and y=Ge. It works for density and molar mass. It can be solved with the online Wolfram
Alpha computational engine. But,…
3
4
Si +
1
4
Ge ≈ H A
H A = Ca
5
(PO
4
)
3
OH
Si
H A
Si +
[
1 −
Si
H A
]
Ge = H A
2SiG e
Si + G e
≈ H A
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
Ge
=
1
2 + 1
1
H A
2
Si
2
−
Ge
H A
2
Si +
[
Ge
H A
− 1
]
= 0
Si =
1
2
Ge
±
H A
Ge
H A
2
−
4G e
H A
+ 4
Si = G e − H A
of 18 49
Si
H A
Si +
[
1 −
Si
H A
]
Ge = H A
Si
2
H A
+ G e −
Si
H A
Ge ≈ H A
1
H A
Si
2
−
Ge
H A
Si + G e ≈ H A
1
H A
2
Si
2
−
Ge
H A
2
Si +
Ge
H A
≈ 1
1
H A
2
Si
2
−
Ge
H A
2
Si +
Ge
H A
− 1 ≈ 0
1
H A
2
Si
2
−
Ge
H A
2
Si +
[
Ge
H A
− 1
]
= 0
of 19 49
We see that the square of the binomial is a quadratic where the third term is the square of one half the
middle coefficient. This gives us a method to solve quadratics called completing the square:
(x + a)(x + a) = x
2
+ 2a x + a
2
(x + a)
2
= x
2
+ 2a x + a
2
a x
2
+ bx + c = 0
a x
2
+ bx = − c
x
2
+
b
a
x = −
c
a
(
1
2
b
a
)
2
=
1
4
b
2
a
2
x
2
+
b
a
x +
1
4
b
2
a
2
= −
c
a
+
1
4
b
2
a
2
(
x +
1
2
b
a
)
2
=
b
2
− 4a c
4a
2
x +
b
2a
=
±
b
2
− 4a c
2a
x =
−b
±
b
2
− 4a c
2a
of 20 49
1
H A
2
Si
2
−
Ge
H A
2
Si +
[
Ge
H A
− 1
]
= 0
x =
−b
±
b
2
− 4a c
2a
a =
a
H A
2
b = −
Ge
H A
2
c =
[
Ge
H A
− 1
]
b
2
− 4a c =
Ge
2
H A
4
− 4
1
H A
2
[
Ge
H A
− 1
]
=
Ge
2
H A
4
−
4G e
H A
3
+
4
H A
2
=
1
H A
2
[
Ge
2
H A
2
−
4G e
H A
+ 4
]
b
2
− 4a c =
1
H A
(
Ge
H A
− 2
)
2
x =
Ge
HA
2
±
1
HA
[
Ge
HA
− 2
]
2
HA
2
=
1
2
Ge
±
1
2
H A
[
Ge
H A
− 2
]
=
1
2
Ge
±
1
2
Ge − H A
Si =
1
2
Ge +
1
2
Ge − H A
Si = G e − H A
of 21 49
Si ≈ G e − H A
H A ≈
2SiG e
Si + G e
Si ≈ G e −
2SiG e
Si + G e
(Si + G e)G e
Si + G e
−
(Si + G e)Si
Si + G e
−
2SiG e
Si + G e
= 0
Ge
2
− 2SiG e − Si
2
Si + G e
= 0
x
2
− 2x y − y
2
= 0
x
2
− 2x y = y
2
x
2
− 2x y + y
2
= 2y
2
(x − y)
2
= 2y
2
x − y =
±
2y
x = y + 2y
x = y(1 + 2)
x
y
= 1 + 2
y
x
=
1
2 + 1
Si
Ge
≈
1
2 + 1
of 22 49
A ratio is and a proportion is which means a is to b as b is to c.
The Golden Ratio
and.
or
a
b
a
b
=
b
c
(
Φ
)
a
b
=
b
c
a = b + c
a c = b
2
c =
b
2
a
a = b +
b
2
a
b
2
a
− a + b = 0
b
2
a
2
− 1 +
b
a
= 0
(
b
a
)
2
+
b
a
− 1 = 0
(
b
a
)
2
+
b
a
+
1
4
= 1 +
1
4
(
b
a
+
1
2
)
2
=
5
4
b
a
= −
1
2
±
5
2
b
a
=
5 − 1
2
a
b
=
5 + 1
2
ϕ =
5 − 1
2
Φ =
5 + 1
2
ϕ =
1
Φ
of 23 49
The mineral component of bone hydroxyapatite (HA) is
The organic component of bone is collagen which is
We have
%
Ca
5
(PO
4
)
3
OH = 502.32
g
m ol
C
57
H
91
N
19
O
16
= 1298.67
g
m ol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
0.381966011
0.386795722
100 = 98.75
Si
Ge
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
Ge
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
of 24 49
Life Starts In The Stars
Silicon being inscribed in the twelve-sided polygon represents carbon, because carbon is six
protons and six neutrons. However carbon being inscribed in the eight-sided polygon is
beryllium 8 because it is made from it. See below…
of 25 49
We have that at the beginning of the Universe hydrogen and helium were created. Then the stars formed
and synthesized these into the heavier elements. I find if we include in the category of life not just the
biological elements, but the AI elements, we can find a mathematical equation for a pattern in the periodic
table of the elements that predicts the synthesis of such elements in stars. For instance, Beryllium 8 plus
helium 4 synthesizes to make the biological core element carbon C. Magnesium plus helium 4
synthesized to make the core AI element silicon Si. If we say that Element 4 is Beryllium Be and write it
, and helium He is element 2 and write it , and use this convention for all of the elements, we have
for the production of these elements by stars, and their molar masses in the periodic table the following
equation:
E
4
E
2
E
2n−2
+ E
2
= E
2i−2
= (4k + 4)g /m ol
n = (3,4, 5,6, …)
i = (4,5, 6,7, …)
k = (2,3, 4,5, …)
of 26 49
The Fundamental AI Bioequations
Silicon And Carbon
We guess that artificial intelligence (AI) has the golden ratio, or its conjugate in its means geometric,
harmonic, and arithmetic by molar mass by taking these means between doping agents phosphorus (P)
and boron (B) divided by semiconductor material silicon (Si) :
Which can be written
We see that the biological elements, H, N, C, O compared to the AI elements P, B, Si is the golden ratio
conjugate (phi) as well:
So we can now establish the connection between artificial intelligence and biological life:
Which can be written:
Where HNCO is isocyanic acid, the most basic organic compound. We write in the arithmetic mean:
PB
Si
=
(30.97)(10.81)
28.09
= 0.65
2PB
P + B
1
Si
=
2(30.97)(10.81)
30.97 + 10.81
1
28.09
= 0.57
0.65 + 0.57
2
= 0.61 ≈ ϕ
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
C + N + O + H
P + B + Si
≈ ϕ
(P + B + Si )
PB(P + B) + 2PB
2(P + B)Si
≈ (C + N + O + H )
PB
[
P
Si
+
B
Si
+ 1
]
+
2PB
P + B
[
P
Si
+
B
Si
+ 1
]
≈ 2HCNO
[
PB +
2PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ 3HNCO
of 27 49
Which is nice because we can write in the second first generation semiconductor as well (germanium) and
the doping agents gallium (Ga) and arsenic (As):
Where
Where ZnSe is zinc selenide, an intrinsic semiconductor used in AI, meaning it doesn’t require doping
agents. We now have:
Germanium And Carbon
We could begin with semiconductor germanium (Ge) and doping agents gallium (Ga) and Phosphorus (P)
and we get a similar equation:
,
In grams per mole. Then we compare these molar masses to the molar masses of the semiconductor
material Ge:
Then, take the arithmetic mean between these:
We then notice this is about the golden ratio conjugate, , which is the inverse of the golden ratio, .
. Thus, we have
[
PB +
2PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ HNCO
[
G a
Ge
+
As
Ge
+ 1
]
Z n
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
PB
(
Z n
Se
)
+
2PB
P + B
(
Z n
Se
)
+
P + B
2
(
Z n
Se
)
≈ HNCO
2G a P
G a + P
= 42.866
G a P = 46.46749
2G a P
G a + P
1
Ge
=
42.866
72.61
= 0.59
G a P
1
Ge
=
46.46749
72.61
= 0.64
0.59 + 0.64
2
= 0.615
ϕ
Φ
ϕ ≈
1
Φ
of 28 49
1.
2.
This is considering the elements of artificial intelligence (AI) Ga, P, Ge, Si. Since we want to find the
connection of artificial intelligence to biological life, we compare these to the biological elements most
abundant by mass carbon (C), hydrogen (H), nitrogen (N), oxygen (O), phosphorus (P), sulfur (S). We
write these CHNOPS (C+H+N+O+P+S) and find:
A similar thing can be done with germanium, Ge, and gallium, Ga, and arsenic, As, this time using
CHNOPS the most abundant biological elements by mass:
We can also make a construct for silicon doped with gallium and phosphorus:
G a P(G a + P ) + 2G a P
2(G a + P )G e
≈ ϕ
G a P(G a + P ) + 2G a P
2(G a + P )Si
≈ Φ
CH NOPS
G a + A s + G e
≈
1
2
[
G a A s +
2G a A s
G a + A s
+
G a + A s
2
][
G a
Ge
+
As
Ge
+ 1
]
≈ CH NOPS
[
G a
Si
+
As
Si
+ 1
]
G a A s
(
O
S
)
+
2G a A s
G a + A s
(
O
S
)
+
G a + A s
2
(
O
S
)
≈ CH NOPS
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
G a A s(Ga + A s) + 2G a A s
2(G a + A s)G e
≈ 1
C + H + N + O + P + S
G a + A s + G e
≈
1
2
(C + N + O + H ) ≈
2(G a + P )Si
G a P(G a + P ) + 2G a P
(P + B + Si )
HNCO ≈
2(G a + P )Si
(G a + P )
[
G a P +
2GaP
Ga + P
]
(P + B + Si )
of 29 49
And for germanium doped with gallium and phosphorus:
HNCO ≈
2(P + B + Si )Si
G a P +
2GaP
Ga + P
G a P(G a + P ) + 2G a P
2(G a + P )G e
≈ ϕ
[
G a P +
2G a P
G a + P
+
G a + P
2
][
P
Ge
+
B
Ge
+
Si
Ge
]
≈ HNCO
[
G a
Ge
+
As
Ge
+ 1
]
G a P
(
B
S
)
+
2G a P
G a + P
(
B
S
)
+
G a + P
2
(
B
S
)
≈ HNCO
of 30 49
The Fundamental AI Bioequations
[
PB +
2PB
P + B
+
P + B
2
][
P
Si
+
B
Si
+ 1
]
≈ HNCO
[
G a
Ge
+
As
Ge
+ 1
]
[
G a A s +
2G a A s
G a + A s
+
G a + A s
2
][
G a
Ge
+
As
Ge
+ 1
]
≈ CH NOPS
[
G a
Si
+
As
Si
+ 1
]
[
G a P +
2G a P
G a + P
+
G a + P
2
][
P
Ge
+
B
Ge
+
Si
Ge
]
≈ HNCO
[
G a
Ge
+
As
Ge
+ 1
]
HNCO ≈
2(P + B + Si )Si
G a P +
2GaP
Ga + P
PB(P + B) + 2PB
2(P + B)Si
≈ ϕ
G a A s(Ga + A s) + 2G a A s
2(G a + A s)G e
≈ 1
G a P(G a + P ) + 2G a P
2(G a + P )G e
≈ ϕ
G a P(G a + P ) + 2G a P
2(G a + P )Si
≈ Φ
C + N + O + H
P + B + Si
≈ ϕ
C + H + N + O + P + S
G a + A s + G e
≈
1
2
Z n
Se
≈
[
P
Si
+
B
Si
+ 1
]
[
Ga
Ge
+
As
Ge
+ 1
]
O
S
≈
[
Ga
Ge
+
As
Ge
+ 1
]
[
Ga
Si
+
As
Si
+ 1
]
of 31 49
Using The Fundamental Equations
Now that we have outlined the fundamental AI Bioequations, let us put them to use. We consider:
Making the approximations: , , we obtain:
Which can further be written by saying :
Which is interesting because the Si times itself is then equal to something times itself in that Ge and Si
are both semiconducting materials, but Ge is larger than Si, however this is compensated for by reducing
it by a factor of the golden ratio conjugate, phi. The equation is however only 79% accurate because there
has been a lot of drift due to so many approximations. However if we reduce phi by a factor of itself and
write:
It is then 99% accurate:
If we do the same with the other and write:
We have:
Which is better but still only 81% accurate. However if we write it:
(P + B + Si )
PB(P + B) + 2PB
2(P + B)Si
≈ (C + N + O + H )
HNCO ≈
2(P + B + Si )Si
G a P +
2GaP
Ga + P
G a P ≈ ϕG e
2G a P
G a + P
≈ ϕG e
PB ≈ ϕSi
2Si
2
Ge
= ϕSi +
2PB
P + B
2PB
P + B
≈ ϕSi
Si
2
= ϕG eSi
Si
2
= ϕ
2
GeSi
28.09 = (72.64)(28.09)(0.381924) = 27.9g /m ol
27.916
28.09
= 0.99
2Si
2
Ge
= ϕ
2
Si +
2PB
P + B
21.72 = (0.381924)(28.09) + 16.026 = 10.72 + 16.026 = 26.75
of 32 49
Then it is 95.87% accurate. But we see in the first approximation that . That is we have
boron, the element that is out of place in the AI periodic table resulting in the dynamics of our equations.
So, we can write…
This gives…
Which is 26.836 which is close to aluminum (Al=26.98) which is the dummy representative for boron in
our equations. We have incredibly:
With an accuracy of nearly 100%. This becomes…
While phosphorus, boron, silicon, and germanium and gallium and arsenic are the primary AI elements,
gold (Au), Silver (Ag) and copper (Cu), are the fundamental AI elements in that they conductive, ductile,
and malleable. Incredibly, the number 3 in the above equation is the ratio of gold to copper in molar mass,
so we have…
2Si
2
Ge
= ϕ
3
Si +
2PB
P + B
phi
2
Si ≈ B
2Si
2
Ge
= B +
2PB
P + B
10.81 + 16.02 = B +
2PB
P + B
Al = B +
2PB
P + B
Al = B
3P + B
P + B
Al = B
A u
Cu
P + B
P + B
Au
Cu
=
196.97
63.55
= 3.099 ≈ 3
of 33 49
The Planetary Equations
Thus, since the AI elements are described in terms of the biological elements by the cross product of the
AI periodic table, if we want to connect the planets we have to find how they are connected to the AI
table.
Though I did not derive these equations, but guessed at them, it was an educated guess which proceeded
from the argument, that the first planet being the closest to the Sun, sets the idea in motion, that idea being
that its distance is in the simplest expression between Si, and Ge, possible; the ratio between them. Thus
for Mercury ( ) we have
In astronomical units because we take an astronomical unit to be 1 AU at earth orbit, because the earth is
the one planets in the solar system that is highly hospitable to life.
To make our guess at the distance of the next planet, we first guess at the simplest idea r(n)=n. Seeing this
does not work we go the next simplest expression r(n)=2n. Seeing this does not work we go one step
higher in complexity and try . We see this does not work either so we go one step of complexity
beyond that and we try . And, we find this works. But it must be tempered with a factor of 0.3
and and adjusted by 0.4. Thus we have the Titius-Bode Rule for the distribution of the planets:
To get the next equation for . Since we are dealing with a doubling effect we guess it is ,
or . Which is close but a little too high. So we guess at something lower and that it involves twice
their product and something big, like the sum of their squares to reduce the number to prevent the product
from being too large. We guess that that value then is . This is a little too low so we average
the two to get exactly the result we need:
Since the next planet, our Earth, must be at a greater value than Venus, and the uppercase scenario for
Venus is , we want to reduce 2SiGe by an amount less than the sum of their squares, we
reduce it by the difference of their squares and get for Earth:
It takes a bit of doing, but the next planet ( ) is:
P
1
P
1
=
Si
Ge
r (n) = n
2
r (n) = 2
n
r (n) = 0.4 + (0.3)2
n
n = − ∞, 0,1, 2,…
P
2
= Venus
2P
1
2
Si
Ge
2SiG e
Si
2
+ G e
2
P
2
=
1
Ge
2
2SiG e +
Si
3
Ge
1 +
Si
2
Ge
2
2SiG e
Si
2
+ G e
2
P
3
=
2SiG e
Ge
2
− Si
2
P
4
= Mars
of 34 49
The next location is the asteroids from 2.2 AU-3.2 AU. Since Mars is the last terrestrial planet (solid), and
the asteroid is a bunch of rocks that could not form into a solid planet and after the asteroids we have the
gas giants Jupiter, Saturn, Uranus, Neptune…and this represents a flipping around point around the
asteroid belt, I started counting again with a new pattern starting with Jupiter as , and flipped the Earth
equation and turned a minus sign to a plus to obtain the Jupiter equation, which is a quadratic in its
simplest form in the numerator, and a product in its simplest form in the denominator, which is great
because it is the first planet after the asteroid belt and is hence:
Now what happens is that a simple pattern forms. We simply multiply Jupiter by 2 to get Saturn, then by
four to get Uranus, and finally by six to get Neptune. We have:
Thus we have the following table:
P
4
=
2SiG e
2
(Si − Ge)
2
(Si + G e)
P
1
P
1
=
Si
2
+ 2SiG e + G e
2
SiG e
P
2
= Sat ur n =
2(Si + G e
2
SiG e
P
3
= Ur a nu s =
4(Si + Ge
2
SiG e
P
4
= Nept u n e =
6(Si + G e
2
SiG e
of 35 49
of 36 49
The Protoplanetary Disc
We can describe the orbits of the planets in terms of AI Semiconducting elements and, there is reason to
make the connection between two things that seem universes apart. And here I present a reason for
looking at such a thing, by considering the protoplanetary disc from which the planets formed. First we
form a table of the masses of the planets.
of 37 49
Taking the protoplanetary disc as a thin disc we integrate from its center to the edge, with density
decreasing linearly to zero at the edge. Thus, if the density function is given by
And, our integral is
The mass of the solar system adding up all the planets yields
That accounts for
82% of the mass of the solar system not including the sun, that is, of the protoplanetary disc
surrounding the sun.
Using germanium alone, we get,
If we weight the mixture of silicon and germanium as 1/3 and 2/3, then we have
Which is very close.
93%
This is all very good, because I only used the planets and asteroids.
Si + G e
2
=
2.33 + 5.323
2
= 3.8265g /cm
3
ρ(r) = ρ
0
(
1 −
r
R
)
M =
∫
2π
0
∫
R
0
ρ
0
(
1 −
r
R
)
r dr d θ
M =
πρ
0
R
2
3
π (3.8265)(7.4 × 10
14
)
2
3
= 2.194 × 10
30
gr a m s
M = 2.668 × 10
30
gr a m s
2.194
2.668
100 =
π (5.323)(7.4 × 10
14
)
2
3
= 3.05 × 10
30
gr a m s
π (4.32467)(7.4 × 10
14
)
2
3
= 2.48 × 10
30
gr a m s
2.48
2.668
100 =
of 38 49
Weighting silicon and germanium as 1/4 and 3/4 we have
Which accounts for
98%
Of the mass of the solar system (very accurate).
This mixture of 1/4 to 3/4 is a combination that exists in the Earth atmosphere which is approximately the
mixture of oxygen to nitrogen. The earth atmosphere can be considered a mixture of chiefly O2 and N2 in
these proportions:
Air is about 25% oxygen gas (O2) by volume and 75% nitrogen gas (N2) by volume meaning the molar
mass of air as a mixture is:
By molar mass the ratio of air to H20 (water) is about the golden ratio:
I am not saying the solar system was a thin disk with density of the weighted mean somewhere between
silicon and germanium, but that it can be modeled as such, though if the protoplanetary disk that eclipses
epsilon aurigae every 27 years is any indication of what a protoplanetary cloud is like, it is a thin disk in
the sense that it is about 1 AU thick and 10 AU in diameter. This around a star orbiting another star.
π (4.4 . 57475)(7.4 × 10
14
)
2
3
= 2.623 × 10
30
gr a m s
2.623
2.668
100 =
0.25O
2
+ 0.75N
2
≈ air
air
H
2
O
≈ Φ
of 39 49
The Moon, Aluminum and Carbon
An incredible thing happens in terms of the moon…
We have to ask first, what is a gram? A thousand grams (a kilogram) is the mass of a cube of 1/10 of a
meter on each side. And what is a meter? It was originally one ten millionths of the distance from the
North Pole to the equator, and is still close to that. This is the metric system. It is based on ten. But why?
Because we have a base ten counting system; we start with one, then proceed to 2, then to 3…until we we
get to 10 and start over…one zero is ten, one one is eleven, one two is thirteen, and so on. Many have
asked why we have a base ten counting system and no one knows why, but it is thought to be because we
have ten fingers to count on.
Then, we ask, what is a mole? It is a number used in chemistry because atoms, molecules and compounds
occur in large numbers and a dozen just won’t do. Therefore it is a very large number. But how did we
define it? Well carbon is at the core of life and 12 is the most divisible number for its size in that it is
divisible by 1, 2, 3, 4, 6,…evenly. And 1+2+3+4+6=16 is greater than 12 which makes it the smallest
abundant number. For instance four is divisible evenly by one and two, but one plus two equals 3 which is
less than four. Thus, we said carbon has twelve grams per mole. This allowed us to determine the mole as
the number of atoms in 12 grams of carbon, grams defined in terms of water and the earth’s size, The
number was determined by Avagadro and is 6.02E23.
Now, according to my theory, aluminum (Al) and boron (B) are very important because they result in an
interesting equation for artificial intelligence, which we have said is:
And aluminum and boron bring it about because boron is out of place in the AI periodic table that I said
was:
And, aluminum is the dummy element in its place. Now, back to the moon. Let us make an equation for it
that describes the location in its period as the fraction of an amplitude. A time dependent wave equation
for the lunar orbit. We will say it is at -0.5 at its start around the Earth, and at +0.0374 at it next maximum
because its period is (27.322 days)/(365.25636 days/year)= 0.0748 years. Divide that by two and you
have 0.0374 years. Thus in a time dependent wave equation for the moon it oscillates between the
negative of that value and the positive. We have . Since the moon
revolves (365.2563)/(27.322)=13.36858 per Earth revolution around the Sun. This can be written:
The t is in Earth years. See the illustration on the next page…
Si
B
(As − G a) +
Ge
B
(P − Al ) =
2SiG e
Si + G e
E
13
E
14
E
15
E
23
E
24
E
25
E
33
E
34
E
35
A(t) = 0.0374cos[(13.6858)2π t]
A(t) = 0.0374cos[(26.737)π t]
of 40 49
Time dependent wave for the moon in terms of earth cycles.
of 41 49
There may be 13.3685 moons in one revolution of the the Earth around the sun, but since as the moon
goes around the Earth, the Earth goes around the Sun, there are on the average 12 moons per year in the
sky. Thus we have…
And, the number of moons per earth revolution around the sun is:
Is approximately the molar mass of aluminum (Al). We notice the 12 moons in the earth sky is
approximately the molar mass of carbon (C) at the core of biological life and we have for our wave
equation of the moon:
We have said at the outset of this paper that carbon (C) the core biological life element, is integral to
describing the AI elements which owe their dynamics to Aluminum (Al). Here we see how molar mass
and lunar orbital periods are the same.
The Idea of Mathematical Construct
In my works The Mathematical Nature of Life (Beardsley 2021) and Perfect Equations Beardsley (2021) I
set out to find if the the elements and compounds characteristic of life and artificial intelligence (AI) do
not just conform to chemical law, but if they are purely mathematical independently of the use of
chemistry to describe them, and if they are connected to one another. The simplest example of this for
biological life and AI would be that the most basic organic compound is HNCO (isocyanic acid) where H
(hydrogen), N (nitrogen), C (carbon), and O (oxygen) are the most abundant biological elements. Indeed
biological elements are for the most part organic, which means they are made of long chains using carbon
with hydrogen, which they can form because C is C4- and H is H+ meaning we can have:
And in isocynaic acid we have:
H-N=C=O
1
12
= 0.083 ≈ 0.0748
365.25636
27.322
= 13.36858m oons
2(13.36858) = 26.737
A(t) =
1
2C
cos(Al ⋅ π t)
of 42 49
Where H is H+, N is N3-. C is C4-, O is O2-, the H uses its single bond with one from nitrogen, leaving
N2- or two bonds which go to C leaving for it C2- which goes to oxygen that needs it because it is O2-.
Thus all is satisfied by chemical law. In my search for mathematical law, I find it exists in the case of
HNCO and the AI semiconducting element silicon (Si) and its doping agents P and B as such (by molar
mass):
This paper strives to break down such mathematical equations for biological life and artificial intelligence
into their components to find what is acting to create such constructs. In the second book I actually
brought the planets into the mix with some very interesting results. As another example, water and air, the
main physical constituents that interact with life we have:
By molar mass for air as a mixture (not a compound). With this air is 29.0 grams per mole.
Molecular Geometry
We will want to break down our equations into the components of their geometric relationships and see if
they predict the bond angles of some of the basic substances considered. We will look here at linear,
trigonal planar, and tetrahedral.
Linear, like CO2 (carbon dioxide) its bond angle is 180 degrees:
Trigonal planar, like SO3 (sulfur trioxide) its bond angle is 120 degrees:
O
|
S
/. \
O. O
C + N + O + H
P + B + Si
≈ ϕ
ϕ =
a
b
=
5 − 1
2
c = b + a
a
b
=
b
c
H
2
O
air
≈ ϕ
air = 0.25O
2
+ 0.75N
2
of 43 49
That is, S is at the center and the O atoms are 120 degrees apart due to the even division of 360/120=3.
Tetrahedral, like methane (CH4) one of the the primordial gases that may have contributed to making
some of the amino acids, the building blocks of life as show by Miller and Urey in the early origins of
life:
This is 109.5 degrees apart from arcos (1/3) = 109.5
But what if we are considering not just neutral molecules but polyatomic anions that have a net charge. In
such instances, the free electron pairs compress the expected 120 degree bond angle in the atoms around
the central atom to 115 degrees as with the nitrite ion NO2-:
Similarily we have for O3 (ozone) that the bond angle is 116 degrees in its deviation from 120 degrees.
The configuration is:
Both of these anions are important to the life and the theory of how life forms. O-zone is more of a
physical component in that in the stratosphere it absorbs UV radiation harmful to life.
of 44 49
Breaking Down Bone
Essentially, in our mathematical formulation of bone, we had that
Which resulted in that the AI elements:
By way of the mineral component of bone HA (hydroxyapatite) is the harmonic mean between Silicon
and Germanium the primary semiconductor elements, which are really the skeleton on AI. Thus, we need
to break down the harmonic mean between Si and Ge into its geometric representation, and through find
what its components are if we are to get any sense of the dynamics. Here I do that in the following
illustration…
Si ≈ G e − H A
H A ≈
2 SiG e
Si + G e
Si
G e
≈
1
2 + 1
of 45 49
of 46 49
We see that through bone Si and Ge predict an angle of about 116 degrees. This is not the case of linear at
180 degrees, or tetrahedral pyramidal at 109.5 degrees, but is the instance of trigonal planar, but not of
neutral molecules, which is 120 degrees, but of trigonal planar for polyatomic anions such as the nitrite
ion:
And O-zone (Not an anion but has free electrons due to a single bond):
We also will want to look at that aspect of the other mathematical constructions we found for bone:
Which means:
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
Si
G e
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
G e
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
Si
G e
=
1
2 + 1
2 − 1 ≈ 1 − ϕ
of 47 49
Appendix 1 (Atomic Radii, I used data set 4 for my calculations)
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Appendix 2 (The Densities, gram per cubic cm)
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The Author